Method of fabricating a shape-changeable magnetic member, method of producing a shape changeable magnetic member and shape changeable magnetic member

ABSTRACT

A method of fabricating a shape-changeable magnetic member comprising a plurality of segments with each segment being able to be magnetized with a desired magnitude and orientation of magnetization, to a method of producing a shape changeable magnetic member composed of a plurality of segments and to a shape changeable magnetic member.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a divisional application of U.S. patent applicationSer. No. 15/596,635, filed May 16, 2017, which claims priority toEuropean Patent Application No. 16175341.3, filed on Jun. 20, 2016, eachof which are incorporated herein by reference in their respectiveentireties.

SUMMARY

The present invention relates to a method of fabricating ashape-changeable magnetic member comprising a plurality of segments witheach segment being able to be magnetized with a desired magnitude andorientation of magnetization, to a method of producing a shapechangeable magnetic member composed of a plurality of segments and to ashape changeable magnetic member.

Shape changeable material is an active material that can control itsgeometry to potentially achieve mechanical functionalities unattainableby traditional machines. Unfortunately, these materials can typicallyonly transform into one or two shapes when their overall dimensionsreach ˜1 cm or smaller on a short time scale.

Other materials are known which can transform into three or four shapes,however, this change is brought about on a comparatively long time scaleof minutes or longer.

Yet other materials are known that can transform into two or more basicshapes that only enable gripping or crawling movements to be realized.

In view of the foregoing it is an object of the present invention tomake available a method of producing small scale shape changeablematerials of ˜1 cm or smaller that can vary their shape very fast with ahigh spatial and temporal resolution. It is a further object of theinvention to make available a method of producing such small scale shapechanging materials in a reproducible and cost effective manner.

These objects are respectively satisfied by a method of fabricating ashape-changeable magnetic member by a method of producing a shapechangeable magnetic member composed of a plurality of segments and by ashape changeable magnetic member as respectively defined in theindependent claims.

According to a first aspect the present invention relates to a method offabricating a shape-changeable magnetic member comprising a plurality ofsegments with each segment being able to be magnetized with a desiredmagnitude and orientation of magnetization, the method comprising thesteps of:

defining a first shape of the member;

defining at least one second shape the member can adopt through theapplication of at least one magnetic field;

determining a desired orientation and magnitude of magnetization foreach of the plurality of segments, such that the member can at leastapproximately change its shape from the first shape to the at least onesecond shape through the application of the at least one magnetic field;and

producing the member by forming each segment such that it at leastsubstantially has the respective desired orientation and magnitude ofmagnetization when the member adopts the first shape in the absence ofthe at least one magnetic field.

The invention thus not only advantageously makes available thepossibility of providing each segment with a differing magnitude ofmagnetic field, but in contrast to what was previously possible alsopermits a control of the orientation of the magnetic field.

Previously the magnitude of the magnetizable material was constant andthus could not be set. Moreover, the orientation of the respectiveadjacent segments was also predefined, so that it was not possible foreach segment to be magnetized with a desired magnitude and orientationof magnetization which could vary from segment to segment in a desirableway.

In particular it was not possible to control the magnitude andorientation of magnetization along a length of the member.

A general methodology is made available by means of the invention toprovide a variety of different shape changeable magnetic members thatcan be used in both static and in dynamic applications through theapplication of different kinds of magnetic fields. In this way verycomplex shapes and also multiple kinds of different shapes can beadopted by the member.

This is because the method permits the member to be programmed so thatthe shapes adopted in the magnetic field correspond to a physicalfunction, so that by applying different fields different functions canbe carried out.

This further permits the use of the shape changeable magnetic materialas e.g. an artificial cilium that could mimic the complex beatingpatterns of their biological counterparts, a spermatozoid-likeundulating swimmer, miniature scale devices such as robots etc.

Advantageously the method further comprises the step of defining betweentwo to twenty, preferably two to ten shapes the member can adopt throughthe application of two to twenty preferably two to ten respectivemagnetic fields differing from one another to bring about a change ofshape between the different shapes. Preferably the change in shape takesplace in seconds or less, more particularly the change in shape can bebrought about at a frequency of 100 Hz.

Preferably the magnetic field can have a constant field strength or agradient, where the gradient is selected such that the shape-changeablemagnetic member can propagate by changing its shape, i.e. to form aspermazotoid like movement thereof.

It is preferred if the desired orientation and magnitude ofmagnetization for at least one shape of the member is determined bydetermining the desired elastic deformations of a member at a spacing salong a length L from an end of the member having a plurality ofsegments of length ds, at a time t and at a point of deflection θ(s, t),the member having a constant cross-sectional area A, by solving thefollowing iterative equation:

${{\tau_{m} + {\int_{s}^{L}{F_{y}\ d\; s\mspace{11mu} \cos \; \theta}} - {\int_{s}^{L}{F_{x}d\; s\mspace{11mu} \sin \mspace{11mu} \theta}}} = {{- \frac{EI}{A}}\frac{\partial^{2}\theta}{\partial\; s^{2}}}},$

where E, and I represent Young's modulus and a second moment of area,respectively, where the applied magnetic torques and forces on theelement are represented by |_(m)(s, t), and F_(x)(s, t) and F_(y)(s, t),respectively, and where the directions of the magnetic forces isindicated by their subscript (x or y).

Using the solution of such an iterative equation n permits thefabrication of members having a constant cross-section and constanttorsional stiffness over their length, i.e. the manufacture ofrectangular shaped members.

It is further preferred if the desired orientation and magnitude ofmagnetisation for at least one shape of the member is determined bydetermining the desired elastic deformations of the member at a spacings along a length L from an end of the member having a plurality ofsegments of length ds, at a time t and at a point of deflection θ(s, t),using the following iterative equation for the member having varyingcross-sectional size A and/or varying moments of inertia I:

${{{\tau_{m}\left( {s,t} \right)}{A(s)}} + {\left\lbrack {\int_{s}^{L}{{F_{y}\ \left( {s,t} \right)}{A(s)}{ds}}} \right\rbrack \cos \; \theta} - {\left\lbrack {\int_{s}^{L}{{F_{x}\ \left( {s,t} \right)}{A(s)}{ds}}} \right\rbrack \mspace{11mu} \sin \mspace{11mu} \theta}} = {{- {{EI}(s)}}\frac{\partial^{2}\theta}{\partial\; s^{2}}\left( {s,t} \right)}$

where E, I, L, and A represent Young's modulus and a second moment ofarea, respectively, where the applied magnetic torques and forces on theelement are represented by □_(m)(s, t), and F_(x)(s, t) and F_(y)(s, t),respectively, and where the directions of the magnetic forces areindicated by their respective subscript (x or y).

Solving such an iterative equation advantageously permits themanufacture of members that are non-uniform in their shape or in theirtorsional stiffness. For example, heart valves typically comprise 3leaflets that together in the closed state approximately form the shapeof a circle, thus in use as e.g. a heart valve a shape changing magneticmember can be provided that has an approximately triangular shape.

Preferably the step of producing the member includes the step ofproviding at least one mold into which a first mixture composed of atleast an elastic material and a magnetizable material is introduced,with the shape of the mold defining the magnitude of the magnetizationof the shape-changeable magnetic member or with the member being cutfrom a material formed in a mold to have a desired shape that can havethe defined magnitude of the magnetization of the shape-changeablemagnetic member or with a shape of the member being printed by use of a3D printer.

In this way the magnitude of the magnetization of the magnetic membercan be tailored to the specific application. This is because eachsegment can be provided with a different amount of magnetizable materialwhich then corresponds to the final magnitude of magnetization.

Advantageously the step of producing the member includes the step ofpermanently magnetising the shape-changeable magnetic member in amagnetic field sufficient to bring about a permanent magnetization.

By permanently magnetizing the member it is ensured that the membermaintains its shape changing capabilities.

In this connection it should be noted that the magnetic field applied topermanently magnetize the member depends on the magnetizable materialselected but typically lies in the region of 1 to 3 T, whereas themagnetic fields applied to bring about the changes of shape of themember typically lie in the region of 1 mT to 100 mT.]

It is preferred if the step of permanently magnetizing theshape-changeable magnetic member is carried out using at least one jighaving a pre-defined shape that subsequently defines the orientation ofmagnetization of the shape-changeable magnetic member.

The jig ensures that orientation of magnetization can at leastapproximately be set in the desired way. This is achieved by selectingthe shape of the jig such that on permanently magnetizing the membereach segment is aligned relative to that magnetic field, so that themagnetizable material aligns with the magnetic field applied.

Advantageously the shape-changeable magnetic member comprises at leasttwo components selected from the group of components consisting of anelastic material, a magnetizable material, a filler material, a metallicfiller material and combinations of the foregoing.

In this way different kinds of materials can be selected as the materialof the magnetic member, so that this can e.g. have a constant stiffnesswith varying magnitude of magnetization a member comprising a elasticmaterial, a magnetizable material, and a metallic filler material areselected.

Preferably the steps of defining the at least first and second shapes ofthe members is carried out to define a motion of the shape-changeablemagnetic member when subjected, at least intermittently, to the at leastone magnetic field.

Thus by applying a magnetic field whose field strength and/ororientation varies a shape changing magnetic member can be madeavailable that is a dynamic member, i.e. it can move through a medium,e.g. like a spermatozoid-like undulating swimmer or a mini-robotcomprising e.g. a camera that is inserted into the human body, e.g. in ablood vessel or into the intestine, and that can then scan structurespresent in the human body to which it is guided through the applicationof a magnetic field.

In a further aspect the present invention relates to a method ofproducing a shape changeable magnetic member composed of a plurality ofsegments, with the segments each having a desired orientation andmagnitude of magnetization, optionally in accordance with at least oneof the preceding claims, the method comprising the steps of:

providing a member composed of at least a first mixture of materialcomprising an elastic material and a magnetizable material;

wherein the member either has the first shape or is supplemented by asecond mixture of material, to form the first shape, with the secondmixture of material comprising at least an elastic material, preferablythe elastic material, and a filler material;

placing the member into at least one jig having a predefined shape; and

magnetizing the member in the jig with a magnetic field sufficient tobring about a permanent magnetization of the plurality of segments.

The jig ensures that orientation of magnetization can at leastapproximately be set in the desired way. This is achieved by selectingthe shape of the jig such that on permanently magnetizing the membereach segment is aligned relative to that magnetic field, so that themagnetizable material aligns with the magnetic field applied.

It should be noted that the advantages described above in connectionwith the method of fabricating a shape-changeable magnetic memberlikewise hold true in connection with this method.

It should also be noted that the member composed of the first mixturecan be formed by introducing the first mixture into a mold or by cuttingmaterial of the first mixture from a rectangular member or by simply 3Dprinting said member.

It should also be noted that this separate method claims can also be adependent step of the method of fabricating a shape-changeable magneticmember.

In this connection it is preferred if the shape of the jig is determinedby the following equation:

$\quad\left\{ \begin{matrix}{{x_{jig}(s)} = {\int_{0}^{s}{{\cos \ \left( {- {\varphi (s)}} \right)}d\; s}}} \\{{y_{jig}(s)} = {\int_{0}^{s}{{\sin \ \left( {- {\varphi (s)}} \right)}d\; s}}}\end{matrix} \right.$

where s is the length of the member formed in the jig, x and y are theshape of the jig in the x and y directions,ϕ(s)=tan⁻¹(m_(y)(s)/m_(x)(s)), and m_(x) and m_(y) represent the x-axisand y-axis components of the magnetization profile, respectively.

In this way the best possible jig can be designed for the member so thatthis can adopt the different shapes and be provided with the desiredorientation of magnetization at least approximately.

Preferably, if the jig has intersecting points, a multi-part jig is usedto form individual segments of the shape changeable magnetic member.

In this way extremely complex members can be produced.

Preferably the individual segments are magnetized separately andsubsequently bonded to one another, wherein the bonding together of theindividual segments can be effected by using a bonding agent that e.g.comprises a mixture of Ecoflex and Aluminum, with the mixture having amass ratio that is at least substantially similar to at least a massratio of the first mixture and optionally of a mass ratio of the firstand second mixture.

Using this kind of mixture a permanent connection between the individualsegments can be ensured.

In a further aspect the present invention relates to a shape changeablemagnetic member that can be obtained by a method in accordance with thedescription provided herein, the member having a plurality of segmentseach having a respective desired orientation and magnitude ofmagnetisation, the member at least comprising an elastic material and amagnetisable material.

The advantages described in the foregoing in relation to the variousmethods also hold true for the shape changeable magnetic member.

Providing such a shape changeable magnetic member allows the member todynamically change its shape when subjected to a magnetic field.

Preferably the magnitude of magnitisation is different for at least someof the plurality of segments.

Alternatively or additionally the orientation of magnetisationadvantageously varies between adjacent segments.

It is preferred if the shape changeable magnetic member comprises afiller material. Such a filler material preferably has the same densityand/or weight as the material of the shape changing member that has theshape changing properties. Moreover, this filler material can beinserted into the member to complement an outer and/or inner shapethereof. For example, it can be used to transform a shape changeablemagnetic member of undulating outer shape into a shape changeablemagnetic member having a rectangular shape.

Further embodiments of the invention are described in the followingdescription of the Figures. The invention will be explained in thefollowing in detail by means of embodiments and with reference to thedrawing in which is shown:

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1a, 1b (i) and 1 b(ii) show the computational methodology and asimple proof-of-concept in order to prepare a shape changing member;

FIGS. 2a, 2b, 2c, 2d, 2e , 2 f show the programming steps required toprogram soft composite materials that can gradually fold up into asemi-circle;

FIGS. 3a, 3b, 3c (i), 3 c(ii), 3 d(i) and 3 d(ii) show the programmingsteps associated with programming a spermatozoid-like undulating softswimmer;

FIGS. 4a (i), 4 a(ii), 4 b, 4 c(i), 4 c(ii), 4 d(i) and 4 d(ii) show thesteps required to program an artificial cilium;

FIGS. 5a, 5b, 5c and 5d show the programming steps and results requiredto create desirable time-varying shapes;

FIGS. 6a, 6b, 6c, 6d, 6e and 6f show the fabrication procedure to createa programmable magnetic soft composite beam;

FIG. 7 the magnetization process of a soft beam;

FIG. 8 shows a tensile test of a mixture of Ecoflex and aluminum withdifferent volume ratios of the aluminum powder;

FIG. 9 an electromagnetic coil system with eight coils that was used togenerate the external magnetic field, B(t);

FIGS. 10a, 10b and 10c show strategies for steering an untetheredprogrammable soft active device images of steps carried out in the glovemold method; and

FIGS. 11a, 11b, 11c and 11d show a further kind of phantom comprising atransition region between a first region and a second region.

DETAILED DESCRIPTION

Any statements made having regard to the direction of a component aremade relative to the position shown in the drawing and can naturallyvary in the actual position of application.

FIG. 1 shows the computational methodology and a simple proof-of-conceptin order to prepare a shape changing member. FIG. 1a shows thecomputational methodology used to magnetically program soft elastomericcomposite materials with complex time-varying shapes (see FIG. 5a forthe detailed computational methodology steps). This concept isillustrated with a straight beam that can be programmed to achieve thedesired shapes shown on the left. The proposed approach uses numericalsimulations to automatically determine the necessary magnetizationprofile, m(s), and magnetic field control inputs, B(t), for the material(shown on the right). The given m(s) and B(t) are only used as anillustration.

FIG. 1b shows a simple proof-of-concept of the proposed method in whicha beam is programmed to create a shape resembling a cosine function whenit is subjected to a 5 mT uniform magnetic field input. FIG. 1b (i)shows the desired shape, and simulated first derivative of the bendingmoment and necessary magnetization profile along the beam. The simulatedfirst derivative of the bending moment plot has two overlapping curves.The lower curve represents the desired first derivative of the bendingmoment required to create the desired shape, and the upper red curverepresents the first derivative of the bending moment generated bymagnetic actuation. The plotted magnetization profile is along thepre-deformed beam (see FIG. 11 for a more quantitative representationfor the magnetization). The obtained experimental results are shown in(ii). The slightly lower line represents the desired programmed shapefor this demonstration. The beam achieved its programmed shape when itwas subjected to a 5 mT magnetic field.

FIG. 2 shows the steps required to program soft composite materials thatcan gradually fold up into a semi-circle. FIG. 2a shows a schematic of asoft beam programmed to fold up under magnetic excitation. Although thismotion was illustrated with only four shapes, there were a total of 100distinct shapes throughout this motion.

FIG. 2b shows the optimized results for the desired first derivative ofthe bending moment. Each plot represents the desired first derivative ofthe bending moment of the beam for one time frame. The frame number foreach time frame is represented by the number at the top of it. In thesimulations, the time difference between each time frame is 0.01seconds. The full lines in the time frames represent the desired firstderivative of the bending moment, and the dotted lines represent theobtained first derivative of the bending moment created by the magneticactuation. The x-axis of each plot represents the length of the beam,which ranges from s=0 mm to 7 mm.

FIG. 2c shows the required magnetization profile, m(s), and the magneticfield, B(t), to achieve the desired time-varying shapes. Thismagnetization profile is along the pre-deformed straight beam (see FIG.11 for a more quantitative representation for the magnetizationprofile). Using the coordinate system in a) as a reference, thevariables B_(x) and B_(y) in the magnetic field plot represent the x-and y-axis components of the magnetic field, respectively.

FIG. 2d shows snapshots of a single beam curling up under magneticexcitation. The yellow lines represent the corresponding desiredtime-varying shapes.

FIG. 2e shows four soft beams made of the programmable material areshown deforming into a reversible “CMU” logo under magnetic excitation.To visualize the logo better, the final “CMU” shape was highlighted withdotted red lines.

FIG. 2f shows a jellyfish-like robot equipped with two soft tentaclesmade of the programmable soft composite material. The robot could propelitself on an oil-water interface by bending its tentacles back and forthunder magnetic excitation.

FIG. 3 shows the programming steps associated with programming aspermatozoid-like undulating soft swimmer. FIG. 3a shows the desiredundulation, which requires a traveling wave with increasing amplitudefrom the left tip to the right tip. The entire motion can be dividedinto two strokes: (i) downward motion and (ii) upward motion. Theassociated time frame for each shape is represented by a correspondingframe number. In the simulations, the time difference between each timeframe is 0.1 seconds.

FIG. 3b shows the optimized results for the desired first-derivative ofthe bending moment to achieve the undulation. Each plot represents thedesired first derivative of the bending moment of the beam for one timeframe. The frame number is represented by the number at the top. Theblue lines represent the desired first derivative of the bending moment,and the dotted red lines represent the obtained first derivative of thebending moment created by the magnetic actuation. The x-axis for eachframe corresponds to the length of the beam, which ranges from s=0 to 10mm.

FIG. 3c shows the required magnetization profile and magnetic field forthe swimmer. This is the magnetization profile along the pre-deformedbeam (see FIG. 11 for a more quantitative representation for themagnetization profile). Using the coordinate system in a as a reference,the variables B_(x) and B_(y) in the magnetic field plot represent thex-axis and y-axis components of the magnetic field, respectively. FIG.3d shows snapshots extracted from the video of the (i) tethered and (ii)untethered undulating swimmer.

FIG. 4 shows the steps required to program an artificial cilium. FIG. 4ashows the extracted two-dimensional natural cilia motion as expressed inCartesian coordinates. The motion pattern includes two strokes: (i) therecovery stroke and (ii) the power stroke. The key time frames used bythe artificial cilium are associated with a corresponding frame number.The time difference between each time frame is 0.2 seconds.

FIG. 4b shows optimized results for the desired first-derivative of thebending moment to achieve the cilium motion. Each plot represents thedesired first derivative of the bending moment of the beam for one timeframe. The frame number is represented by the number at the top of it.The blue lines represent the desired first derivative of the bendingmoment, and the dotted red lines represent the obtained first derivativeof the bending moment created by the magnetic actuation. The x-axis foreach frame corresponds to the length of the beam, ranging from s=0 to 10mm. The frames in the middle of the motion were given more weight duringthe optimization process because they were more important.

FIG. 4c (i) shows the required magnetization profile and FIG. 4c (ii)shows the magnetic field and its spatial gradients for the cilium (seeFIG. 11 for a more quantitative representation for the magnetizationprofile). Using the coordinate system in FIG. 4a as a reference, thevariables B_(x) and B_(y) in the magnetic field plot represent thex-axis and y-axis components of the magnetic field, respectively. Thespatial gradients B_(xx), B_(xy), and B_(yy) represent

$\frac{\partial B_{x}}{\partial x},{\frac{\partial B_{y}}{\partial x}\mspace{14mu} {and}\mspace{14mu} \frac{\partial B_{y}}{\partial y}},$

respectively, FIG. 4d shows snapshots extracted from the video of thebeating artificial cilium.

FIG. 5a shows the programming steps required to create desirabletime-varying shapes. FIG. 5b shows the computational results based onthe desired kinematics in FIG. 5b (i), a quasi-static analysis can beconducted on FIG. 5b (ii).

FIG. 5c shows the conditions used to obey the boundary conditions. For afixed end, there should not be any deflections. By contrast, thereshould be no bending moment at the free end.

FIG. 5d shows the creation of an artificial extension to satisfy theboundary conditions for the free-ends. At all of the time instants, thebending moment at the free end should be zero, as achieved byintroducing an artificial extended free-end (indicated by the redportions). The bending moment along this artificial extension was fittedby a polynomial function so that its bending moment at the free end wasalways zero.

FIG. 6 shows the fabrication procedure to create a programmable magneticsoft composite beam. FIG. 6a shows a negative mold for the beam. Thismold is formed like a rectangle to form a rectangular shaped magnetizedmember.

In a first step the mold is filled with a passive component in order toform a rectangular shaped member. FIG. 6b shows the mold filled with thepassive component. A mixture of Al and Ecoflex was used as the passivecomponent and simply poured into the mold in liquid form and allowed tocure.

Having calculated the ideal shape required to form a magnetized memberof desired magnitude and orientation, see e.g. equation 1, a negativemold for the magnetized component is formed in the passive component.FIG. 6c shows the step of removing the material of passive component toform a non-uniform width in the passive component by cutting out thematerial with a laser cutter. The shape of the band of non-uniform widthis based on the programmed magnetization profile.

In a further step the magnetizable material, i.e. an active component,is filled into the negative mold formed in the passive component. FIG.6d shows the active component, comprising a mixture of NdFeB andEcoflex, that was poured into the mold formed by the passive componentand cured.

FIG. 6e shows the installation of the beam in a jig and how the beam wasbent to fit into the jig's profile. The profile of the jig is based oncalculations that enable the formation of a jig that can be used to formthe shape changeable magnetized member. Equation 21 can be iterativelysolved to design the magnetization profile that is to be introduced intothe member by means of the jig and a source of magnetization.

FIG. 6f shows the step of magnetizing the beam with a strong B-field (˜1T) in the jig. In order to magnetize the member this is placed adjacentto electric coils such as those shown in FIG. 9. Due to the shape of thejig, the member can be magnetized with a specific profile. Havingprovided a member with active material of non-uniform shape (see FIG. 6d) this is also magnetized with a non-uniform magnitude of magnetization.

FIG. 7 shows the magnetization process of a soft beam. The soft beam wasplaced in a jig, which was designed based on the magnetization profilefrom the simulation results. A large external constant uniform magneticfield was applied in the +x direction. The soft beam was thenmagnetically programmed by the applied magnetic field but due to itselastic nature can still recover to a straight shape after being removedfrom the jig.

FIG. 8 shows a tensile test of the mixture of Ecoflex and aluminum withdifferent volume ratios of the aluminum powder. The data for the mixtureof Ecoflex and aluminum is represented as circles, and the data for themixture of Ecoflex and NdFeB are represented as squares. A linear modelwas fitted with the parameters shown in the figure.

FIG. 9 shows an electromagnetic coil system with eight coils that wasused to generate the external magnetic field, B(t). In the experiments,the shape-programmable materials were placed in a container filled withwater or oil, and the container was in turn placed in the center of theworkspace of the electromagnetic coil system. The time-varying shapes ofthe beams were recorded by the camera.

FIG. 10 shows strategies for steering an untethered programmable softactive device. Each coordinate frame represents the device's body frame.FIG. 10a shows that by designing the tentacles of the jellyfish-likerobot to be symmetrical about its body frame's y-axis, the netmagnetization was designed to always be parallel to its body frame'sy-axis. By changing the direction of B, the orientation of the robot wasaligned with the field.

FIG. 10b shows that the material is constrained on a water surface ifthe member is placed into water. The magnetic torque created by m_(z)was counterbalanced by the surface tension of the water. The orientationof the material was controlled by applying a B in the x-y plane, wherethe m_(x) of the net magnetization would align with the applied B.

FIG. 10c shows how a rigid component with a programmed magnetizationprofile can be added. By applying a spatial gradient,

$\frac{\partial B_{z}}{\partial x},$

the rigid component experienced a coupled force, F₁, that allowed it torotate about the z-axis.

FIG. 11: Quantitative representation for the magnetization profiles forall the showcases when they were un-deformed. a. Magnetization profilefor the cosine showcase. b. Magnetization profile for the jellyfish-likerobot and reversible “CMU” logo. c. Magnetization profile for theartificial cilium. d. Magnetization profile for the spermatozoid-likeundulating swimmer.

Shape-programmable magnetic materials have more potential to createcomplex time-varying shapes than do other shape-programmable materialsthat are controlled by heat, light, chemicals or pressure because themagnetic field control inputs can be specified not only in theirmagnitude but also in their directions. While shape-programming viamagnetic means has the potential to create a new paradigm ofprogrammable matter, this concept is relatively new. The discussion onthis subject in prior literature is limited to fabrication processesthat can create direction-varying magnetization profiles within magneticmaterials, and to custom showcases which rely heavily on humanintuition. The critical programming steps to acquire the necessarymagnetization profile and magnetic field control inputs for complextime-varying shapes have not yet been developed. In view of thesechallenges, a generic computational methodology is presented that canmagnetically program soft elastomeric composite materials to achievefast time-varying shapes (with transformations <1 second) with highspatial and temporal resolutions.

Although this methodology can also be applied to macro-scale devices(see item 1.6 in the following), the discussions presented herein willprimarily focus on miniature devices, which have overall dimensions thatare approximately 1 cm or smaller. Despite being only equipped withsimple functionalities such as rigid-body and gripping motions, existinguntethered miniature devices have already been employed across a widerange of applications pertaining to microfluidics, microfactories,bioengineering, and healthcare. The proposed methodology could thereforeincrease the impact of these devices as it has the potential tosignificantly enhance their locomotion and manipulation capabilities.

A unique advantage of the proposed methodology is that it is notinhibited by the limits of human intuition since it uses numericalsimulations to automatically generate the necessary magnetizationprofile, m, and actuating magnetic fields, B (FIG. 1a ). Thecomputational method was implemented to create desired two-dimensionalshapes in this study for simplicity, but it could be directly extendedto three-dimensional shapes in the future (see item 1.3). For thesimulations pertaining to miniature devices, practical constraints wereconsidered, such as having a time-invariant m and a position-invariantB. These constraints are necessary because it is neither easy tore-magnetize the device when in situ nor easy to use the electromagneticsystem presented herein (see FIG. 9) to create extremely large spatialgradients in the magnetic field to achieve local variation for suchsmall-scale devices.

The programming steps with beams that are subjected to quasi-staticconditions (see FIG. 5a ) are presented in the following. To simplifythe discussion, the beams' cross-sectional area are constrained to beuniform and only allow them to bend in a plane. Following the steps inFIG. 5a , the desired elastic deformations along the beam's length, sare first defined. Because the desired elastic deformations can varywith time, t, the kinematics can be mathematically represented with therotational deflections along the beam, θ(s, t) (FIG. 5b (i)). After thekinematics are specified, the Euler-Bernoulli theory can be applied andquasi-static analysis on an infinitesimal element, ds, at any time, t,to obtain a generic governing equation that is applicable for both smalland large deformations (see FIG. 5b (ii)):

$\begin{matrix}{{{\tau_{m} + {\int_{s}^{L}{F_{y}\ d\; s\mspace{11mu} \cos \; \theta}} - {\int_{s}^{L}{F_{x}d\; s\mspace{11mu} \sin \mspace{11mu} \theta}}} = {{- \frac{EI}{A}}\frac{\partial^{2}\theta}{\partial\; s^{2}}}},} & (1)\end{matrix}$

where E, I, L, and A represent the Young's modulus, the second moment ofarea, the length and the cross-sectional area of the beam, respectively.The applied magnetic torque and forces on the element are represented by□_(m)(s, t), F_(x)(s, t) and F_(y)(s, t), and their values are dictatedby m(s) and B(t). The direction of the magnetic forces, F_(x) and F_(y),is indicated by their subscript. The left-hand side of Eq. (1)represents the external magnetic actuation, and the right-hand siderepresents the required first derivative of bending moment to achievethe desired shape. The right-hand side of the equation can also bewritten as

${{- \frac{1}{A}}\frac{\partial M_{b}}{\partial s}},$

where M_(b) represents the bending moment. Equation (1) implies that thematerial's desired time-varying shapes can be achieved if the magneticactuation can be programmed to balance the required bending momentacross the entire length for all time. To determine the necessary m(s)and B(t) for the desired actuation, these are first represented withgeneral Fourier series. This representation allows both m(s) and B(t) toinclude all possible functions, regardless of whether they arecontinuous or discrete. Subsequently, both m(s) and B(t) areautomatically determined by numerically optimizing their Fourier seriescoefficients to approach the equality in Eq. (1) (see items 1.1-1.2).The material is then magnetized with m(s) by following the stepsdescribed in item 2, allowing it to generate the desired time-varyingshapes when actuated by the obtained B(t).

As the first experimental demonstration of the shape-programmingmethodology, a millimeter-scale silicone rubber beam with embeddedmagnetic and metallic microparticles was programmed to create a singleshape when it was subjected to a constant B (FIG. 1b (i)). By using theobtained m(s) from the presented computational method, it is shown thatthe beam achieved its desired cosine function shape when magneticallyactuated (FIG. 1b (ii)) (See items 2-3 and FIGS. 6-8 for the materialcomposition and fabrication of the beam). Next, the effectiveness of thepresented methodology was demonstrated by programming variousmillimeter-scale beams to create multiple desired shapes. First, acomposite silicone rubber beam was programmed that could achieve simpletime-varying shapes. For this beam, the time-varying shapes were dividedinto 100 discrete time frames. In each time frame, the beam's curvaturewas held constant throughout its length, gradually increasing betweeneach time frame, until the beam curls into a semi-circle (FIG. 2a ).Despite having time-varying shapes, appropriate yet simple m(s) and B(t)were achieved that satisfied Eq. (1) (FIG. 2b-c ). Using the obtainedm(s) and B(t), the beam was experimentally manipulated to form thedesired shapes (FIG. 2d ). Because the required magnetization profileand magnetic fields were relatively simple, this concept was extended tosimultaneously control multiple beams that had similar motions. Byorganizing several such beams in specific orientations, a reversiblethree-letter “CMU” logo (FIG. 2e ) was created. This concept was furtherextended by using two similar beams to form the tentacles of ajellyfish-like robot, which can swim on an oil-water interface. Bycontrolling the time-varying shapes of the tentacles, a power stroke anda recovery stroke are able to be created. When the speed of the powerstroke was greater than that of the recovery stroke, a net propulsionthat allowed the jellyfish-like robot to swim against the slope of theoil-water interface (FIG. 2f and item 1.6 for the discussion of changingthe stroke speeds) was able to be created. The jellyfish-like robot wasalso steerable, and the details of such steering strategies werediscussed in item 1.4 and FIG. 10.

It was also demonstrated that the proposed method could create highlycomplex time-varying shapes. To illustrate this, an undulating swimmerwas created, as well as an artificial cilium that was able toapproximately mimic the complex beating patterns of its biologicalcounterpart. Although the feasibility of creating a magneticallyactuated undulating soft swimmer had been previously demonstrated¹⁶, thetraveling waves along those swimmers were restricted to uniformamplitude. The creation of an undulating swimmer that has a travelingwave with non-uniform amplitudes is more challenging and has not yetbeen demonstrated. Thus, a propagating traveling wave similar to theswimming gait of a spermatozoid was created in this work, in which theamplitudes are gradually increased from the fixed end to the free end(FIG. 3a ). The obtained m(s) and B(t) for creating the desiredundulating swimmer are shown in FIG. 3c . By tethering the undulatingsoft swimmer, the created traveling wave on the swimmer's body (FIG. 3d(i)) was visually illustrated. It was further demonstrated that theuntethered swimmer used its spermatozoid-like undulation to swim on anair-water interface (FIG. 3d (ii)).

Finally, an artificial soft cilium was created that was able toapproximate the required beating pattern of a biological cilium. Thebiological beating pattern was divided into power and recovery strokes(FIG. 4a ). Due to the high complexity of the motion, the solution couldeasily be trapped into a sub-optimal solution if only one optimizationprocess was utilized. Thus, the programming steps for the artificialcilium were segregated into two sequential optimization processes. Thefirst optimization determined the necessary m(s) and B_(rec)(t) for therecovery strokes. The obtained m(s) was subsequently fed into a secondoptimization process to determine the required B_(pow)(t) for the powerstrokes. The obtained results are shown in FIG. 4b-d . The key timeframes and shapes that were utilized to closely mimic the complexbeating pattern of a biological cilium are shown in FIG. 4a . Althoughother researchers have had some success in creating time-asymmetricalmotions for their artificial cilia, the artificial cilium presentedherein is the only one on a millimeter scale that can approximate themotions of a biological cilium.

While the proposed programming method is promising, there are severallimitations that need to be addressed in future studies. First, althoughboth m(s) and B(t) are represented with their corresponding optimalone-dimensional (1D) Fourier series, the obtained magnetic actuationcannot be represented by a two-dimensional (2D) Fourier series in termsof s and t. This fact implies that the proposed method cannot produceall possible shapes when m(s) is time-invariant and B(t) isposition-invariant. However, this limitation may be moderated bydeveloping more powerful electromagnetic systems that can allow B tochange spatially. Theoretically, this moderation will allow thepresented method to produce a larger range of feasible shapes (see item1.6). While a complete analysis to determine the range of feasibleshapes that can be achieved by the presented method is beyond the scopeof this paper, a brief discussion on this topic is provided in item 1.5.Second, several metastable shapes may exist for a given control input,and these shapes may cause the programmable material to deform into anundesired shape. Because the selected metastable shape is highlydependent on the previous shape, this limitation can be moderated byusing a finer temporal resolution for the shape trajectories. Thismoderation reduces the deviation between the desired shape and theprevious shape, making it easier to guide the material to deform intothe desired shape. Third, because the proposed method uses a numericaloptimization approach, the obtained solution may not be the globallyoptimal solution. New numerical techniques, such as the one used for thecilium case, can be used to overcome this limitation. Fourth, while ageneric theory is provided in items 1.1-1.3, these are limited by thecurrently available fabrication techniques from experimentally programmaterials that are smaller than millimeter-scale and also materials withnon-planar three-dimensional geometries.

In this way a systematic methodology was introduced that can guidescientists and engineers to magnetically program desired time-varyingshapes for soft materials. The method was validated with a simpleshowcase, and its effectiveness was demonstrated by creating areversible three-letter logo, a jellyfish-like robot, aspermatozoid-like undulating swimmer, and an artificial cilium. Comparedto other shape-programmable materials that may require minutes to inducea shape change, the presented devices can transform into their desiredshapes within seconds. It is envisioned that this methodology may enableresearchers to develop novel soft programmable active surfaces anddevices that could be utilized in robotics, engineering, materialsscience, and medicine.

In order to determine the desired orientation and magnitude ofmagnetization for at least one shape of the member one has to determinethe desired elastic deformations of a member at a spacing s along alength L from an end of the member having a plurality of segments oflength ds, at a time t and at a point of deflection θ(s, t), the memberhaving a constant cross-sectional area A, this is done by solvingequation 1.

In the following the derivation of this equation will be discussed indetail.

1.1 Derivation of Equations Used to Determine a Desired Orientation andMagnitude of Magnetization for Each of the Plurality of Segments:

In the following the detailed quasi-static analysis and optimizationformulations for programming magnetic soft materials with time-varyingshapes will be presented. In particular, the critical steps to acquirethe necessary magnetization profiles and actuating magnetic fields areprovided. These analyses are valid for beams that have large or smallbending deflections. Although the boundary conditions of the beam arecritical and will be discussed in the next sub-section, the analysiswill start with these boundary conditions: the beam is fixed at s=0 andhas a free end at s=L (FIG. 5b (i)). Using these boundary conditions andwithout any loss in generality, the bending deflections of the beam witha global frame that has its z-axis parallel to the beam's bending axisis described.

Due to the assumption that the shape of the beam is known across alltime frames, the torque balance equation for an arbitrary infinitesimalelement that is shown in FIG. 5b (ii) at a given time, t, can beexpressed as

$\begin{matrix}{{{\tau_{m}A} + {v\mspace{11mu} \cos \mspace{11mu} \theta} - {h\mspace{11mu} \sin \mspace{11mu} \theta}} = {- {\frac{\partial M_{b}}{\partial s}.}}} & (2)\end{matrix}$

The variable Mb represents the beam's bending moment, and h and vcorrespond to the x- and y-axis internal forces within the beam,respectively. In a similar manner, the force balance equations of theinfinitesimal element can be expressed as

$\begin{matrix}{{F_{x} = {{- \frac{1}{A}}\frac{\partial h}{\partial s}}},{F_{y} = {{- \frac{1}{A}}{\frac{\partial v}{\partial s}.}}}} & (3)\end{matrix}$

Thus, by using the Euler-Bernoulli equation and substituting Eq. (3)into Eq. (2), the desired deflections can be expressed explicitly by theactuating magnetic forces and torques as follows:

$\begin{matrix}{{\tau_{m} + {\int_{s}^{L}{F_{y}\ d\; s\mspace{11mu} \cos \; \theta}} - {\int_{s}^{L}{F_{x}d\; s\mspace{11mu} \sin \mspace{11mu} \theta}}} = {{- \frac{EI}{A}}\frac{\partial^{2}\theta}{\partial\; s^{2}}}} & (4)\end{matrix}$

The applied magnetic forces and torques are dictated by themagnetization profile and actuating magnetic fields, and theirrelationships can be mathematically described as

$\begin{matrix}\left\{ \begin{matrix}{\tau_{m} = {\begin{bmatrix}0 & 0 & 1\end{bmatrix}\left\{ {{R\left( {s,t} \right)}{m(s)} \times {B(t)}} \right\}}} \\{F_{x} = {\begin{bmatrix}1 & 0 & 0\end{bmatrix}\left\{ {\left\lbrack {{R\left( {s,t} \right)}{m(s)}} \right\rbrack \cdot \nabla} \right\} {B(t)}}} \\{F_{y} = {\begin{bmatrix}0 & 1 & 0\end{bmatrix}\left\{ {\left\lbrack {{R\left( {s,t} \right)}{m(s)}} \right\rbrack \cdot \nabla} \right\} {B(t)}}}\end{matrix} \right. & (5)\end{matrix}$

The rotational matrix, R(s, t), is used to account for the orientationchange in the magnetization profile due to the beam's large deflection,and it is given as

$\begin{matrix}{R = \begin{pmatrix}{\cos \mspace{11mu} \theta} & {{- \sin}\mspace{11mu} \theta} & 0 \\{\sin \mspace{11mu} \theta} & {\cos \mspace{11mu} \theta} & 0 \\0 & 0 & 1\end{pmatrix}} & (6)\end{matrix}$

In contrast to other magnetic programming studies, human intuition toderive the necessary m(s) and B(t) is not used. Rather computers areused to automatically generate these parameters. This is achieved byfirst representing them with a Fourier series:

$\begin{matrix}{{{m(s)} = \begin{bmatrix}{\sum\limits_{i = 0}^{n}\left\{ {{a_{i}\; {\cos \left( {i\; \omega_{s}s} \right)}} + {b_{i}\; {\sin \left( {i\; \omega_{s}s} \right)}}} \right\}} \\{\sum\limits_{i = 0}^{n}\left\{ {{c_{i}\; {\cos \left( {i\; \omega_{s}s} \right)}} + {d_{i}\; {\sin \left( {i\; \omega_{s}s} \right)}}} \right\}} \\0\end{bmatrix}},{{B(t)} = \begin{bmatrix}{\sum\limits_{j = 0}^{n}\left\{ {{\alpha_{j}\; {\cos \left( {j\; \omega_{t}t} \right)}} + {\beta_{j}\; {\sin \left( {j\; \omega_{t}t} \right)}}} \right\}} \\{\sum\limits_{j = 0}^{n}\left\{ {{\gamma_{j}\; {\cos \left( {j\; \omega_{t}t} \right)}} + {\eta_{j}\; {\sin \left( {j\; \omega_{t}t} \right)}}} \right\}} \\0\end{bmatrix}},{\begin{bmatrix}\frac{\partial B_{x}}{\partial x} \\\frac{\partial B_{y}}{\partial x} \\\frac{\partial B_{y}}{\partial y}\end{bmatrix} = {\begin{bmatrix}{\sum\limits_{j = 0}^{m}\left\{ {{\epsilon_{j}\; {\cos \left( {j\; \omega_{t}t} \right)}} + {\delta_{j}\; {\sin \left( {j\; \omega_{t}t} \right)}}} \right\}} \\{\sum\limits_{j = 0}^{m}\left\{ {{\lambda_{j}\; {\cos \left( {j\; \omega_{t}t} \right)}} + {\mu_{j}\; {\sin \left( {j\; \omega_{t}t} \right)}}} \right\}} \\{\sum\limits_{j = 0}^{m}\left\{ {{\rho_{j}\; {\cos \left( {j\; \omega_{t}t} \right)}} + {\sigma_{j}\; {\sin \left( {j\; \omega_{t}t} \right)}}} \right\}}\end{bmatrix}.}}} & (7)\end{matrix}$

Thus, by substituting the fitting functions in Eq. (7) into Eq. (1),this can be rewritten as

$\begin{matrix}{{\sum\limits_{j = 0}^{n}{\sum\limits_{j = 0}^{m}\left\{ {{\left\lbrack {{\alpha_{i}\gamma_{i}} - {c_{i}\alpha_{j}}} \right\rbrack \cos \; \theta \; \cos \; \left( {i\; \omega_{s}s} \right){\cos \left( {j\; \omega_{t}t} \right)}} + \ldots - {\left\lbrack {{d_{i}\eta_{j}} + {b_{i}\beta_{j}}} \right\rbrack \sin \; \theta \; \sin \; \left( {i\; \omega_{s}s} \right){\sin \left( {j\; \omega_{t}t} \right)}}} \right\}}} + {\cos \; \theta {\sum\limits_{i = 0}^{n}{\sum\limits_{j = 0}^{m}\left\{ {{\left\lbrack {{a_{i}\lambda_{j}} + {c_{i}\rho_{j}}} \right\rbrack {\int_{s}^{L}{\cos \; \theta \; {\cos \left( {i\; \omega_{s}s} \right)}{\cos \left( {j\; \omega_{t}t} \right)}d\; s}}} + \ldots - {\left\lbrack {{b_{i}\sigma_{j}} - {d_{i}\mu_{j}}} \right\rbrack {\int_{s}^{L}{\sin \; {{\theta sin}\left( {i\; \omega_{s}s} \right)}{\sin \left( {j\; \omega_{t}t} \right)}{ds}}}}} \right\}}}} - {\sin \; \theta {\sum\limits_{i = 0}^{n}{\sum\limits_{j = 0}^{m}\left\{ {{{{\left\lbrack {{a_{i}\epsilon_{j}} + {c_{i}\lambda_{i}}} \right\rbrack {\int_{s}^{L}{\cos \; \theta \; {\cos \left( {i\; \omega_{s}s} \right)}{\cos \left( {j\; \omega_{t}t} \right)}{ds}}}} + \ldots - \left. \quad{\left\lbrack {{b_{i}\mu_{j}} - {d_{i}\delta_{j}}} \right\rbrack {\int_{s}^{L}{\sin \; {{\theta sin}\left( {i\; \omega_{s}s} \right)}{\sin \left( {j\; \omega_{t}t} \right)}d\; s}}} \right\}} = {{- \frac{EI}{A}}\frac{\partial^{2}\theta}{\partial s^{2}}}},} \right.}}}} & (8)\end{matrix}$

where the left-hand side of the equation is a linear combination of theproducts of the Fourier coefficients. A computational approach is usedto determine the ideal values of the Fourier coefficients so Eq. (8) canbe satisfied. First, the motion of the beam is discretized into p numberof time frames, i.e., t=t₀; t=t₁; . . . ; t=t_(p). Similarly, in eachtime frame, the length of the beam is divided into q number of segments,i.e., s=s₀; s=s₁; . . . ; s=s_(q)=L. Thus, q new equations for each timeframe are created by substituting different values of s across theentire beam length into Eq. (8). By assembling all of the equationsacross all time frames, there are a total of p×q equations that arelinearly dependent on the products of the one-dimensional Fouriercoefficients. This can be written in matrix form as

Ku=M _(b).  (9)

Subsequently, the ideal Fourier coefficients are solved by performingthe following optimization process:

minimize f=(Ku−M _(b))^(T) Q(Ku−M _(b))

subjected to: |m(s)|≤m _(max),

|B(t)|≤B _(max)  (10)

where Q is a matrix that gives different weightings/importance todifferent time frames/shapes. The time frames/shapes that are deemed tobe more important have higher weightings. Physically, the optimizationprocess in Eq. (10) minimizes the difference/errors between the magneticactuation and the required first derivative of the bending moment withinthe beam. This optimization process is solved numerically by usingsolvers, such as a genetic algorithm and gradient-based solvers. Moreinformation pertaining to the nature of these solvers can be found in.

After the optimization solvers determined the ideal Fouriercoefficients, the necessary m(s) and B(t) required to achieve thedesired time-varying shapes were determined. The benefit of usingFourier series to represent m(s) and B(t) is that they are inclusive ofall possible mathematical functions. Thus, this enables the proposedprogramming method to be highly versatile, and it can be used as ageneric approach.

1.2 Boundary Conditions

To achieve the desired shapes, the boundary conditions of the beams mustalways be satisfied. Generally, there are two types of boundaryconditions: the fixed and free ends of the beam (FIG. 5c ). For thefixed ends of the beams, the desired kinematics, such as the verticaland rotational deflections, must be zero at all time-instants. However,because it is difficult to create rigid-body torques at the free ends,the bending moment at the free ends must be constrained to zero.However, because not all time-varying shapes have a zero bending momentat their free end, a method that can overcome this limitation isintroduced (see FIG. 5d ).

The proposed method is to extend the desired length of the beamartificially. Subsequently, the bending moment of the beam of theartificial extension is gradually reduced to zero by using a polynomialfitting curve (FIG. 5d ). The necessary magnetization profile for thisextension can be determined to satisfy the free-end boundary conditionby including the artificial extension into the Fourier seriesrepresentation of m(s). This method was used to program the cosinefunction, jellyfish-like robot and the spermatozoid-like undulatingswimmer. It is not necessary to utilize this method for the artificialcilium because its desired motions have automatically satisfied thefree-end boundary condition.

1.3 Programming Materials with Non-Planar Shapes

Similar to the formulations for programming beams, the desiredtime-varying displacement fields (u, v, w) for the materials isinitially specified. Based on these deflections, the time-varyingdeformation gradient tensor, F, and strains can be computed. Theresultant time-varying strains across the materials are given as

$\begin{matrix}\left\{ \begin{matrix}{e_{ij} = {\frac{1}{2}\left( {u_{i,j} + u_{j,i}} \right)}} \\{\omega_{ij} = {\frac{1}{2}\left( {u_{i,j} - u_{j,i}} \right)}} \\{ɛ_{i,j} = {e_{ij} + {\frac{1}{2}\left( {e_{ki} + \omega_{ki}} \right)\left( {e_{kj} + \omega_{kj}} \right)}}}\end{matrix} \right. & (11)\end{matrix}$

where ε_(ij) is the component of the Lagrangian strain tensor and thesubscripts indicate the Cartesian directions, i.e., i, j, k=1, 2, 3. Theabove equations are written in index notation for convenience. Therelationship between the strains and stresses within the materials canbe expressed by

S=Cε  (12)

where S is the stress tensor in Lagrangian description, i.e., the secondPiola-Kirchhoff stress (2^(nd) PK stress), and C is a 9×9 matrix of thecoefficients determined by the material properties.

After the stress distribution within the materials is determined, thequasi-static analysis is established. As it is more desirable to performthe quasi-static analysis in Eulerian description, the 2^(nd)-PK stressis converted into the Cauchy stress in the Eulerian description as

σ=J ⁻¹ F·S·F ⁷  (13)

where J=det(F). According to a theory for electromagnetic-elasticsolids, the quasi-static equations can be written in index notation as

$\begin{matrix}\left\{ \begin{matrix}{{\sigma_{{kl},l} + f_{l}} = 0} \\{{\left( {\sigma_{kl} - \sigma_{lk}} \right) + \tau_{i}} = 0}\end{matrix} \right. & (14)\end{matrix}$

where σ_(ij) is the component of the Cauchy stress tensor f_(i) andτ_(i) and are the external body force and torque per unit volume in thei^(th)-direction, respectively. The external body forces and torques pervolume that are applied from the magnetic field are functions of thelocal magnetization vector, m:

τ_(m) =m×B and f _(m)=(∇B)·m  (15)

When Eq. (15) is expressed in index notation, it becomes

$\begin{matrix}\left\{ \begin{matrix}{f_{i} = {m_{i}B_{i,j}}} \\{\tau_{i} = {{{B_{k}m_{l}} - {B_{l}m_{k}}} = {\epsilon_{kli}m_{l}B_{k}}}}\end{matrix} \right. & (16)\end{matrix}$

Because one must account for the change in the magnetization profileafter the material has deformed, the magnetization profile has to bemapped from its initial un-deformed state to the current deformed state.Because m is defined as the magnetization per unit volume, its magnitudevaries similarly to the density of a body that undergoes a deformation.Thus, m can be defined as

$\begin{matrix}{m = \frac{dM}{dV}} & (17)\end{matrix}$

where dM is magnetic moment within the volume of dV. When a deformationoccurs, the magnetic moment changes its orientation, and the magnitudeof the volume is also changed:

dM′=RdM

dV′=Jdv  (18)

where R is the rotational component of F, which can be found by thepolar decomposition of F. Therefore, the magnetization vector underdeformation can be written as

$\begin{matrix}{m^{\prime} = {\frac{{dM}^{\prime}}{{dV}^{\prime}} = {\frac{RdM}{jdV} = {\frac{R}{J}m}}}} & (19)\end{matrix}$

This implies that the magnetic torque/forces in the deformed state canbe expressed as τ_(m)′=m′×B and f_(m)′=(∇B)·m′. By substituting thesevariables back into Eqs. (14) and (16), the equilibrium equationsexpressed explicitly with the magnetic torques and forces are obtained.To generalize the approach to solve for these magnetic torques andforces, the Fourier series are again used to represent the magnetizationprofiles and magnetic fields. Similar to Eq. (10), an optimizationapproach is then used to determine the optimal Fourier coefficients.This determines the m and B necessary to achieve the time-varying shapesfor a programmable material with non-planar geometries.

1.4 Steering Strategies

There are several strategies to steer untethered miniature devicesmagnetically. The first steering strategy is introduced by using thejellyfish-like robot as an example. To implement this strategy, themagnetization profile of the beams was intentionally constrained to besymmetrical around the y-axis of the robot's body frame (see FIG. 10a ).This allowed the robot's net magnetization to always be parallel to itsbody frame's y-axis. The net magnetization of the programmable material,mid, is given as

m _(net) =A∫ ₀ ^(L) m(s)ds  (20)

Because the net magnetization of the robot always aligns with B(t), thedirections of B(t) can be varied to control the orientation of therobot. Furthermore, because the required B to change the shape of thetentacles is always in the same direction, the shape of the tentacles'can be controlled by adjusting the magnitude of B(t) after the robotachieves its desired orientation.

In addition to the first strategy, two additional strategies arepresented that allow untethered programmable materials to steer in aplane while being able to achieve their desired shape transformations.The second strategy is to constrain certain motions of the programmablematerial so that it is easier to steer the device. The last strategy isto include a rigid component that can be used to control the device'sorientation.

The second strategy can be implemented by placing the material on aliquid interface in which the programmable material is constrained bythe surface tension of the fluid. As an illustration, FIG. 10b shows thex-z plane of the programmable material's body frame. Due to the surfacetension of the fluid, the z-axis components of the net magnetizationcannot create rigid-body torques that affect the orientation of thematerial. Thus, the alignment of the material on the liquid interface issolely dependent on the body frame's x-axis component of the netmagnetization. Thus, the robot's orientation can be controlled by usingan applied B(t) to align this x-axis component of the net magnetization.This strategy was used to control the orientation of the undulatingswimmer.

For the last steering strategy, one can control the orientation of thedevice by programming the magnetization profile of a rigid component.Multiple feasible magnetization profiles may exist, and an example isshown in FIG. 10c . In this case, although the net magnetization for therigid-component is zero, this component can still provide a rigid-bodytorque that can be used to steer the orientation of the material. Byfollowing the body frame assignment in FIG. 10c , a rigid-body torquearound the z-axis can be induced on the rigid component when the spatialgradient,

$\frac{\partial B_{z^{\prime}}}{\partial x}$

is applied, allowing the material to steer in the x-y plane.

This spatial gradient also induces a z-axis force for the x-axiscomponents of m(s), and the induced deflections into the plane can begreatly reduced by increasing the stiffness of the beam in thatdirection. This can be easily achieved by increasing the width of thebeam. Thus, it is possible to compensate for any z-axis torque that isinduced by the programmable material by controlling the magnitude of thespatial gradient,

$\frac{\partial B_{z^{\prime}}}{\partial x}$

1.5 Achievable Time-Varying Shapes

Although a complete analysis for determining the number of feasibleshapes that are achievable with this method is beyond the scope of thisapplication, a brief discussion is presented here. The proposed methodcannot produce all possible time-varying shapes for small-scalematerials because the materials have a time-invariant m and a global Bthat cannot be spatially changed. Thus, the number of programmableshapes for a material depends significantly on the complexity of theshape trajectories. For example, for simple time-varying shapes, such asthose shown in FIG. 2, it is possible to create 100 shapes for theentire shape trajectories. However, for extremely complex time-varyingshapes, such as those generated by the artificial cilium, 5 key shapesfor the device can be planned. Currently, the only way to determine thenumber of achievable shapes is to perform the numerical optimizationprocess shown above at 1.1. If the optimization cannot create thenecessary magnetic actuation to match the desired first derivative ofthe bending moment, the number of shapes must be reduced. This processmay have to be iterated several times until it is possible to obtain agood numerical solution. However, it is believed that the minimum numberof continuous shapes achievable by the proposed method should be twobecause it is possible to pattern two axes of the magnetization profileindependently, i.e., the x- and y-axis components of m(s).

1.6 Additional Discussion

In the following the possibility of extending the proposed approach tosimultaneously determine the magnetization profile for multiple beamsand the effects of a B that can be varied locally in space will bediscussed. Finally, the possibility of changing the speed for the shapetransformations in the experiments is discussed.

The programming method can determine the magnetization profile formultiple beams simultaneously by using multiple sets of the Fourierseries to represent the magnetization profile of these beams. Forexample, if there are r numbers of beams, there will be r sets ofFourier series. Thus, for each time frame, one can create r×q newequations by substituting different values of s across each beam intoEq. (8). By assembling all of the equations across all time frames,there will be a total of p×r×q equations that are linearly dependent onthe products of the one-dimensional Fourier coefficients. Using theformulations shown in Eq. 10, the optimal Fourier coefficients can bedetermined, thus generating the necessary magnetization profiles for allof the beams.

On the other hand, if B can be varied locally for l regions, there willbe l number of independent B values, i.e., there will be B₁, B₂, . . .B_(l). Each of these magnetic fields can then be represented by aFourier series, i.e., there are l sets of them. However, Eq. 8 will beslightly modified as one substitutes the corresponding B in each region.In a similar manner, the optimal Fourier coefficients can be solved byEq. 10. The difference in a B that can be varied locally in space isthat it allows the beam to create more feasible motions. Furthermore, aB that can be varied locally is more practical for materials inmacro-scale because they can be achieved with smaller magnitudes ofspatial gradients. There is no loss in generality to apply the proposedapproach for programming macro-scale devices.

Lastly, the feasibility of changing the speed for inducing the shapechanges is discussed. Physically, there is an upper speed limit for theprogrammable material to change its shape. This limit is dictated byeither the speed of the electromagnetic system that generates theactuating magnetic fields or the fundamental natural frequency of thematerial. In the experiments, it is the speed of the electromagneticsystem that limits the bandwidth to be 25 Hz. Based on this limitation,the fastest component of the Fourier series representing B to be 25 Hzhas been constrained. Reducing the speed for the shape change is,however, much simpler and there is no lower bound for such a change.Thus, for the jellyfish-like robot, the speed of the recovery stroke hasbeen reduced to be approximately 3 times slower than its power stroke.

2. Fabrication Techniques

In the following the fabrication technique to create the desiredmagnetization profile for a programmable beam is discussed. The requiredsteps are illustrated in FIG. 6.

The programmable magnetic soft composite material consists of twocomponents: a passive component and an active component that can bestimulated by magnetic excitation. The active component is created byembedding fine neodymium-iron-boron (NdFeB) particles that have anaverage size of 5 (MQFP, Magnequench) into a soft silicone rubber(Ecoflex 00-10, Smooth-on, Inc.). The volume ratio for the NdFeBparticles and Ecoflex 00-10 is 0.15:1. However, the passive component iscreated by embedding aluminum (Al) powder with an average particle sizeof 5 μm into the same type of silicone rubber with the same volumeratio. The volume ratio of the active and passive components is selectedsuch that their elastic modulus will match (see point 3 below),providing the composite with a uniform elastic modulus. The relationshipbetween the passive component's volume ratio and its resultant elasticmodulus was experimentally characterized (see point 3 in the following).To create a non-uniform magnetization profile that has a varyingmagnitude, the distribution between the passive and active componentsmust be patterned. The locations that have a higher magnitude ofmagnetization will have more active components. To achieve this, atwo-step micro-molding process was adopted. First, a negative mold thathad the desired geometries of the beams was created by computernumerical control machining on an acrylic sheet. The passive component(in liquid form) was first poured into the negative mold and allowed tocure. Once the passive component was fully cured, a laser cutter wasused to cut out a band that had a non-uniform width. Subsequently, theactive component (in liquid form) was poured into the mold to replacethe removed band. The two components formed a composite that had auniform thickness once the active component was also cured. Due to thenon-uniform width of the band, the distribution of the active componentscould be patterned. This allowed the beam to have a magnetizationprofile with a varying magnitude when the beam was magnetized by auniform magnetic field (of approximately 1 T). The orientation of thedesired magnetization profile was created by using jigs to bend/fold thebeam during the magnetization process (FIG. 7). The curvature of thejigs can be mathematically represented by the following integral:

$\begin{matrix}\left\{ \begin{matrix}{{x_{jig}(s)} = {\int_{0}^{s}{{\cos \left( {- {\varphi (s)}} \right)}{ds}}}} \\{{y_{jig}(s)} = {\int_{0}^{s}{{\sin \left( {- {\varphi (s)}} \right)}{ds}}}}\end{matrix} \right. & (21)\end{matrix}$

where □=tan⁻¹(m_(y)(s)=m_(x)(s)), and m_(x) and m_(y) are the x- andy-axis components of m(s) when the beam is undeformed. The desired jigswere fabricated with the laser cutter. Thus, by magnetizing the beamwhen it was sandwiched between the jigs, the desired magnetizationprofile could be obtained after the applied magnetic field and the jigswere removed. The NdFeB particles that were embedded within the activecomponents were saturated by the large magnetizing field.

It should be noted in this connection that if the jig has intersectingpoints, it will be necessary to divide the jig/beam into multiplesegments to prevent such intersections. Thus, each of these segmentswill have to be magnetized separately and subsequently ‘glued’ to reformthe beam. This adhesive then can be glued using e.g. a mixture ofEcoflex+Aluminum, e.g. a material having the same mass ratio as that ofthe member.

3. Experimental Procedures

This section describes the experimental procedures for matching theelastic modulus of the passive and active components. The procedures andsetup for the experiments, which evaluate the performance of theprogrammable materials, are also discussed.

3.1 Matching the Elastic Modulus Properties

Because of the embedded metal particles, the elastic modulus of thecomposite materials is different from that of pure Ecoflex. The embeddedaluminum and NdFeB powders were selected to have the same mean particlessize of 5 μm. The volume ratio of the embedded NdFeB powder to Ecoflexin the active component was predetermined; hence, the component'selastic modulus was fixed. Therefore, the elastic modulus of the passivecomponent, Ecoflex with embedded aluminum powder, was tuned by changingthe volume ratio of the particles to Ecoflex. The elastic modulus ofboth the passive and active components was evaluated with a tensiletesting machine (Instron 5943, Instron Inc.). Each volume ratio wasevaluated with three experiments, and a linear model was fitted torepresent the relationship between the elastic modulus and the volumeratio. Based on the fitted model, the necessary volume ratio for thepassive component's elastic modulus to match the active components wasdetermined (see FIG. 8). The corresponding mass ratio was obtained by

$\begin{matrix}{{{mass}\mspace{14mu} {ratio}} = {\frac{{density}\mspace{14mu} {of}\mspace{14mu} {particle}}{{density}\mspace{14mu} {of}\mspace{14mu} {Ecoflex}} \times {volume}\mspace{14mu} {{ratio}.}}} & (22)\end{matrix}$

3.2 Experimental Procedure

The magnetic field and its spatial gradients were generated by anelectromagnetic coil system with eight coils, as shown in FIG. 9. Thecoil system can be controlled to generate the desired magnetic field andits spatial gradient in the workspace with a uniformity above 95% acrossa 2 cm×2 cm×2 cm volume. The mapping from the current in each coil andthe resulting magnetic field and spatial gradient can be approximated ina linear form as

$\begin{matrix}{{AI} = {\begin{pmatrix}B \\B_{grad}\end{pmatrix}.}} & (23)\end{matrix}$

The matrix A and vector I represent the actuation matrix and thecurrents for each coil, respectively. The magnetic field can beexpressed as B=[B_(x) B_(y) B_(z)]^(T) in the global frames shown in thefigures, and the spatial gradients of B are represented by B_(grad).Based on Gauss's Law of ∇·B=0 and Ampere's Law of ∇×B=0_(3×1), there areonly five independent components. Because there is more than onecombination of B_(grad), selected the following representation forB_(grad) was selected:

$\begin{matrix}{B_{grad} = {\left\lbrack {\frac{\partial B_{x}}{\partial x}\mspace{14mu} \frac{\partial B_{x}}{\partial y}\mspace{14mu} \frac{\partial B_{y}}{\partial y}\mspace{14mu} \frac{\partial B_{z}}{\partial x}\mspace{14mu} \frac{\partial B_{z}}{\partial y}} \right\rbrack^{T}.}} & (24)\end{matrix}$

The shape change using the simulated magnetic field was recorded and itsspatial gradients on the programmable beams.

The governing equation for a non-uniform cross-sectional area, A, and anon-uniform second moment of area, I, across the beam:

${{{\tau_{m}\left( {s,t} \right)}{A(s)}} + {\left\lbrack {\int_{s}^{L}{{F_{y}\left( {s,t} \right)}{A(s)}{ds}}} \right\rbrack \cos \; \theta} - {\left\lbrack {\int_{s}^{L}{{F_{x}\left( {s,t} \right)}{A(s)}{ds}}} \right\rbrack \sin \; \theta}} = {{- {{EI}(s)}}\frac{\partial^{2}\theta}{\partial s^{2}}\left( {s,t} \right)\mspace{11mu} \left. {(*} \right)}$

This equation can also be iteratively solved to produce a beam withnon-uniform E.

3.3 Parameters for Each Showcase:

In the following the parameters are provided that were used for eachshowcase, i.e., the dimensions of the beams and the number of Fourierseries coefficients, or n and m, respectively. These parameters aresummarized in Table 1.

TABLE 1 Parameters for each showcase. Jellyfish-like Undulating Cosinerobot Swimmer Cilium Length (mm) 7 7 10 10 Width (mm) 5 3 3 3 Thickness(μm) 80 80 240 80 m — 10 1 10 n 200 10 70 20

The quantitative representation of each showcases' magnetization profileis shown in FIG. 11.

What is claimed is:
 1. A method of producing a shape changeable magneticmember composed of a plurality of segments, with the segments eachhaving a desired orientation and magnitude of magnetization, the methodcomprising the steps of: providing a member composed of at least a firstmixture of material comprising an elastic material and a magnetizablematerial; wherein the member either has the first shape or issupplemented by a second mixture of material, to form the first shape,with the second mixture of material comprising at least an elasticmaterial, preferably the elastic material, and a filler material;placing the member into at least one jig having a predefined shape; andmagnetizing the member in the jig with a magnetic field sufficient tobring about a permanent magnetization of the plurality of segments. 2.The method of according to claim 1, wherein the plurality of segments ofthe shape-changeable magnetic member are magnetized with a desiredmagnitude and orientation of magnetization, the method comprising thesteps of: defining a first shape of the member; defining at least onesecond shape the member can adopt through the application of at leastone magnetic field; determining a desired orientation and magnitude ofmagnetization for each of the plurality of segments, such that themember can at least approximately change its shape from the first shapeto the at least one second shape through the application of the at leastone magnetic field; and producing the member by forming each segmentsuch that it at least substantially has the respective desiredorientation and magnitude of magnetization when the member adopts thefirst shape in the absence of the at least one magnetic field.
 3. Themethod in accordance with claim 1, wherein the shape of the jig isdetermined by the following equation: $\quad\left\{ \begin{matrix}{{x_{jig}(s)} = {\int_{0}^{s}{{\cos \left( {- {\varphi (s)}} \right)}{ds}}}} \\{{y_{jig}(s)} = {\int_{0}^{s}{{\sin \left( {- {\varphi (s)}} \right)}{ds}}}}\end{matrix} \right.$ where s is the length of the member formed in thejig, x and y are the shape of the jig in the x and y directions,ϕ(s)=tan⁻¹(m_(y)(s)/m_(x)(s)), and m_(x) and m_(y) represent the x-axisand y-axis components of the magnetization profile, respectively.
 4. Themethod in accordance with claim 1, wherein, if the jig has intersectingpoints, a multi-part jig is used to form individual segments of theshape changeable magnetic member.
 5. The method in accordance with claim4, wherein the individual segments are magnetized separately andsubsequently bonded to one another, wherein the bonding together of theindividual segments can be effected by using a bonding agent thatcomprises a mixture of Ecoflex and Aluminum, with the mixture having amass ratio that is at least substantially similar to at least a massratio of the first mixture.
 6. The method in accordance with claim 5,wherein the mass ratio of the mixture is at least substantially similarto a mass ratio of the first and second mixture.
 7. A shape changeablemagnetic member that can be obtained by a method of fabricating ashape-changeable magnetic member comprising a plurality of segments witheach segment being able to be magnetized with a desired magnitude andorientation of magnetization, the method comprising the steps of:defining a first shape of the member; defining at least one second shapethe member can adopt through the application of at least one magneticfield; determining a desired orientation and magnitude of magnetizationfor each of the plurality of segments, such that the member can at leastapproximately change its shape from the first shape to the at least onesecond shape through the application of the at least one magnetic field;and producing the member by forming each segment such that it at leastsubstantially has the respective desired orientation and magnitude ofmagnetization when the member adopts the first shape in the absence ofthe at least one magnetic field, the member having a plurality ofsegments each having a respective desired orientation and magnitude ofmagnetisation, the member at least comprising an elastic material and amagnetisable material.
 8. The shape changeable magnetic member of claim7, wherein the magnitude of magnitisation is different for at least someof the plurality of segments.
 9. The shape changeable magnetic member ofclaim 7, wherein the orientation of magnetisation varies betweenadjacent segments.
 10. The shape changeable magnetic member of claim 7,further comprising a filler material.
 11. A shape changeable magneticmember that can be obtained by a method of producing a shape changeablemagnetic member composed of a plurality of segments, with the segmentseach having a desired orientation and magnitude of magnetization, themethod comprising the steps of: providing a member composed of at leasta first mixture of material comprising an elastic material and amagnetizable material; wherein the member either has the first shape oris supplemented by a second mixture of material, to form the firstshape, with the second mixture of material comprising at least anelastic material, preferably the elastic material, and a fillermaterial; placing the member into at least one jig having a predefinedshape; and magnetizing the member in the jig with a magnetic fieldsufficient to bring about a permanent magnetization of the plurality ofsegments, the member having a plurality of segments each having arespective desired orientation and magnitude of magnetisation, themember at least comprising an elastic material and a magnetisablematerial.
 12. The shape changeable magnetic member of claim 11, whereinthe magnitude of magnitisation is different for at least some of theplurality of segments.